Sparse Kneser graphs are Hamiltonian

For integers k≥1 and n≥2k+1, the Kneser graph K(n,k) is the graph whose vertices are the k-element subsets of {1,…,n} and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form K(2k+1,k) are also known as the odd graphs. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every k≥3, the odd graph K(2k+1,k) has a Hamilton cycle. This and a known conditional result due to Johnson imply that all Kneser graphs of the form K(2k+2a,k) with k≥3 and a≥0 have a Hamilton cycle. We also prove that K(2k+1,k) has at least 22k−6 distinct Hamilton cycles for k≥6. Our proofs are based on a reduction of the Hamiltonicity problem in the odd graph to the problem of finding a spanning tree in a suitably defined hypergraph on Dyck words

Prognostic significance of claudin expression changes in breast cancer with regional lymph node metastasis.

Adherent and tight junction molecules have been described to contribute to carcinogenesis and tumor progression. Additionally, the group of claudin-low tumors have recently been identified as a molecular subgroup of breast carcinoma. In our study, we examined the expression pattern of claudins, beta-catenin and E-cadherin in invasive ductal (IDCs) and lobular (ILCs) carcinomas and their corresponding lymph node metastases (LNMs). Tissue microarrays of 97 breast samples (60 invasive ductal carcinomas, 37 invasive lobular carcinomas) and their corresponding LNMs have been analyzed immunohistochemically for claudin-1, -2, -3, -4, -5, -7, beta-catenin and E-cadherin expression. The stained slides were digitalized with a slide scanner and the reactions were evaluated semiquantitatively. When compared to LNMs, in the IDC group beta-catenin and claudin-2, -3, -4 and -7 protein expression showed different pattern while claudin-1, -2, -3, -4 and -7 were differently expressed in the ILC group. Lymph node metastases developed a notable increase of claudin-5 expression in both groups. Decrease or loss of claudin-1 and expression of claudin-4 in lymph node metastases correlated with reduced disease-free survival in our patients. According to our observations, the expression of epithelial junctional molecules, especially claudins, is different in primary breast carcinomas compared to their lymph node metastases as demonstrated by immunohistochemistry. Loss of claudin junctional molecules might contribute to tumor progression, and certain claudin expression pattern might be of prognostic relevance

Links between oral health and personality at a law enforcement school

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